Complete theory

Remarks

Example: for any $L$-structure $\mathcal{M}$, the theory $\mathrm{Th}(\mathcal{M})$ of all sentences true in $\mathcal{M}$ is complete. It has the model $\mathcal{M}$, and for each sentence $\varphi$ either $\varphi$ or $\neg\varphi$ is true in $\mathcal{M}$, hence lies in $\mathrm{Th}(\mathcal{M})$.

Non-example: the theory of groups has a model but is not complete. For $\varphi = \forall x \, \forall y \, (x \cdot y = y \cdot x)$, the group $\Z$ satisfies $\varphi$ and the symmetric group $S_3$ satisfies $\neg\varphi$.

A satisfiable theory $T$ is complete if and only if all its models are elementarily equivalent. By the completeness theorem, $T \models \varphi$ can be replaced by $T \vdash \varphi$, provability from $T$. Some authors do not require clause 1.